simplipy-real

Real flavour of rules found in both rule files: the rule over the reals, with only the assumptions that make it defined.

Case set `ba9dc4def822`

case	sympy best	sympy-ask	sympy-refine	sympy-simplify	wolfram best	wolfram-fullsimplify	wolfram-refine	wolfram-simplify
sp-acj4-1-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-102-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-108-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	no verdict	true ✓
sp-acj4-120-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-127-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-128-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-131-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-132-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-133-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-136-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-137-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-14-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-145-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-162-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-179-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-181-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-185-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-194-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-195-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-196-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-218-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-22-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-220-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-233-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-248-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-249-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-251-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-252-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-253-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-256-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-275-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-276-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-283-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-286-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-289-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-31-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-326-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-33-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-35-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-354-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-356-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-359-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-364-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-367-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-368-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-369-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-370-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-371-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-372-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-373-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-38-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-394-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-395-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-396-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-397-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-402-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-404-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-412-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-413-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-414-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-415-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-417-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-421-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-5-real contingent	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓	no verdict ✓
sp-acj4-73-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-80-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-83-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-86-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-90-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-99-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓

sp-acj4-1-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \arcsin\left(\left|x\right|\right) \overset{?}{=} \left|\arcsin x\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0 -> abs asin _0

Lean proof: included

(case sp-acj4-1-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0  ->  abs asin _0")
  (var x real)
  (ask (= (asin (abs x)) (abs (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r1w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-102-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)

Lean proof: included

(case sp-acj4-102-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6  ->  abs / _0 (-6)")
  (var x real)
  (ask (= (/ (abs x) 6) (abs (/ x -6))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-108-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{\left|x\right|}{x} = \frac{x}{\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0

Lean proof: included

(case sp-acj4-108-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0  ->  / _0 abs _0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (abs x)))
  (ask (= (/ (abs x) x) (/ x (abs x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; have hr : (r:ℂ) ≠ 0 := h0.2; have ha : ((|r|:ℝ):ℂ) ≠ 0 := (by have := h1.2; rwa [AE.abs_ofReal] at this); simp only [AE.abs_ofReal, AE.inv_def]; rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_eq_div_iff hr ha]; exact_mod_cast (abs_mul_abs_self r)"))

sp-acj4-120-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0

Lean proof: included

(case sp-acj4-120-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0  ->  abs / np.pi _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ pi (abs x)) (abs (/ pi x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, abs_of_pos Real.pi_pos]"))

sp-acj4-127-real

Given

\(x \in \mathbb{R},\quad e^{x} \neq 1,\quad e^{x} \neq -1\)

Claim

\(\displaystyle \left|\operatorname{artanh}\left(e^{x}\right)\right| \overset{?}{=} \operatorname{artanh}\left(e^{x}\right)\)

Expected

contingent true at \(x = -1\); false at \(x = 1\)

equal at some admissible points, different at others

witness (true): \(x = -1\)

witness (false): \(x = 1\)

SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0 -> atanh exp _0

Lean proof: included

(case sp-acj4-127-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0  ->  atanh exp _0")
  (var x real)
  (assume (!= (exp x) 1))
  (assume (!= (exp x) -1))
  (ask (= (abs (atanh (exp x))) (atanh (exp x))))
  (expect contingent)
  (witness true (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w1 | exact AE.w2_exp_neg_one_ne_one | exact AE.w2_exp_neg_one_ne_neg_one | exact AE.w2_atanhD_exp_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w2 | exact AE.w2_exp_one_ne_one | exact AE.w2_exp_one_ne_neg_one | exact AE.w2_atanhD_exp_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-128-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\cos\left(\arcsin x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0 -> cos asin _0

Lean proof: included

(case sp-acj4-128-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (cos (asin x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r128w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-131-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\cosh x\right)\right| = \log\left(\cosh x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0

Lean proof: included

(case sp-acj4-131-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0  ->  log cosh _0")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (abs (log (cosh x))) (log (cosh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h0.2⟩, by simp⟩, trivial, h0.2⟩, ?_⟩; rw [AE.cosh_ofReal, AE.log_ofReal (Real.cosh_pos r).le, AE.abs_ofReal, abs_of_nonneg (Real.log_nonneg (Real.one_le_cosh r))]"))

sp-acj4-132-real

Given

\(x \in \mathbb{R},\quad \sin x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0

Lean proof: included

(case sp-acj4-132-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0  ->  neg log sin _0")
  (var x real)
  (assume (nonzero (sin x)))
  (ask (= (abs (log (sin x))) (- (log (sin x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w1 | exact AE.w2_sin_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w2 | exact AE.w2_sin_neg_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-133-real

Given

\(x \in \mathbb{R},\quad \tanh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0

Lean proof: included

(case sp-acj4-133-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0  ->  neg log tanh _0")
  (var x real)
  (assume (nonzero (tanh x)))
  (ask (= (abs (log (tanh x))) (- (log (tanh x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w1 | exact AE.w2_tanh_one_ne | exact AE.w2_tanhD_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w2 | exact AE.w2_tanh_neg_one_ne | exact AE.w2_tanhD_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-136-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)

Claim

\(\displaystyle \left|x^{e^{1}}\right| \overset{?}{=} x^{e^{1}}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e -> pow _0 np.e

Lean proof: included

(case sp-acj4-136-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e  ->  pow _0 np.e")
  (var x real)
  (assume (or (nonzero x) (> (re (exp 1)) 0)))
  (ask (= (abs (pow x (exp 1))) (pow x (exp 1))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first  | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, Complex.exp_im]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r136w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-137-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)

Claim

\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi

Lean proof: included

(case sp-acj4-137-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi  ->  pow _0 np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (ask (= (abs (pow x pi)) (pow x pi)))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r137w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-14-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\sin x = \sin\left(-x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0 -> sin neg _0

Lean proof: included

(case sp-acj4-14-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0  ->  sin neg _0")
  (var x real)
  (ask (= (- (sin x)) (sin (- x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [Complex.sin_neg]"))

sp-acj4-145-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\sin\left(\arccos x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0 -> cos asin _0

Lean proof: included

(case sp-acj4-145-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (sin (acos x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r145w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-162-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{arcosh}\left(\operatorname{surd}\left(x, 5\right)\right) \overset{?}{=} \operatorname{arcosh}\left(x^{\frac{1}{5}}\right)\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5 -> acosh pow _0 0.2

Lean proof: included

(case sp-acj4-162-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5  ->  acosh pow _0 0.2")
  (var x real)
  (ask (= (acosh (surd x 5)) (acosh (pow x 1/5))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r162w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-179-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arccos x\right) \neq i,\quad \sinh\left(\arccos x\right) \neq -i,\quad \cosh\left(\arccos x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arccos x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arccos x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0 -> asin tanh acos _0

Lean proof: included

(case sp-acj4-179-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0  ->  asin tanh acos _0")
  (var x real)
  (assume (!= (sinh (acos x)) i))
  (assume (!= (sinh (acos x)) (- i)))
  (assume (nonzero (cosh (acos x))))
  (ask (= (atan (sinh (acos x))) (asin (tanh (acos x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w1 | exact AE.w2_r179w1_hI | exact AE.w2_r179w1_hnI | exact AE.w2_r179w1_atanD | exact AE.w2_r179w1_cosh | exact AE.w2_r179w1_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w2 | exact AE.w2_r179_hI | exact AE.w2_r179_hnI | exact AE.w2_r179_atanD | exact AE.w2_r179_cosh | exact AE.w2_r179_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-181-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arcsin x\right) \neq i,\quad \sinh\left(\arcsin x\right) \neq -i,\quad \cosh\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arcsin x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arcsin x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0 -> asin tanh asin _0

Lean proof: included

(case sp-acj4-181-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0  ->  asin tanh asin _0")
  (var x real)
  (assume (!= (sinh (asin x)) i))
  (assume (!= (sinh (asin x)) (- i)))
  (assume (nonzero (cosh (asin x))))
  (ask (= (atan (sinh (asin x))) (asin (tanh (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r181w2 | exact AE.w2_r181_hI | exact AE.w2_r181_hnI | exact AE.w2_r181_atanD | exact AE.w2_r181_cosh | exact AE.w2_r181_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-185-real

Given

\(x \in \mathbb{R},\quad \cos\left(\cos x\right) \neq 0,\quad \tan\left(\cos x\right) \neq i,\quad \tan\left(\cos x\right) \neq -i\)

Claim

\(\displaystyle \arctan\left(\tan\left(\cos x\right)\right) = \cos x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0 -> cos _0

Lean proof: included

(case sp-acj4-185-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0  ->  cos _0")
  (var x real)
  (assume (nonzero (cos (cos x))))
  (assume (!= (tan (cos x)) i))
  (assume (!= (tan (cos x)) (- i)))
  (ask (= (atan (tan (cos x))) (cos x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h1.1.2⟩, by rw [AE.cos_ofReal, AE.tan_ofReal]; exact AE.atan.D_ofReal _⟩, trivial⟩, ?_⟩; rw [AE.cos_ofReal, AE.tan_ofReal, AE.atan_ofReal, Real.arctan_tan (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]) (by linarith [Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-194-real

Given

\(x \in \mathbb{R},\quad \sin\left(\cos x\right) \neq 1,\quad \sin\left(\cos x\right) \neq -1,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\cos x\right)\right) = \operatorname{arsinh}\left(\tan\left(\cos x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0 -> asinh tan cos _0

Lean proof: included

(case sp-acj4-194-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0  ->  asinh tan cos _0")
  (var x real)
  (assume (!= (sin (cos x)) 1))
  (assume (!= (sin (cos x)) -1))
  (assume (nonzero (cos (cos x))))
  (ask (= (atanh (sin (cos x))) (asinh (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.cos r := (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]); have hy2 : Real.cos r < Real.pi / 2 := (by linarith [Real.cos_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.cos_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.cos r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.cos r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.cos_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-195-real

Given

\(x \in \mathbb{R},\quad \sin\left(\sin x\right) \neq 1,\quad \sin\left(\sin x\right) \neq -1,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\sin x\right)\right) = \operatorname{arsinh}\left(\tan\left(\sin x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0 -> asinh tan sin _0

Lean proof: included

(case sp-acj4-195-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0  ->  asinh tan sin _0")
  (var x real)
  (assume (!= (sin (sin x)) 1))
  (assume (!= (sin (sin x)) -1))
  (assume (nonzero (cos (sin x))))
  (ask (= (atanh (sin (sin x))) (asinh (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.sin r := (by linarith [Real.neg_one_le_sin r, Real.pi_gt_three]); have hy2 : Real.sin r < Real.pi / 2 := (by linarith [Real.sin_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.sin_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.sin r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.sin r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.sin_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-196-real

Given

\(x \in \mathbb{R},\quad \sin\left(\tanh x\right) \neq 1,\quad \sin\left(\tanh x\right) \neq -1,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\tanh x\right)\right) = \operatorname{arsinh}\left(\tan\left(\tanh x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0 -> asinh tan tanh _0

Lean proof: included

(case sp-acj4-196-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0  ->  asinh tan tanh _0")
  (var x real)
  (assume (!= (sin (tanh x)) 1))
  (assume (!= (sin (tanh x)) -1))
  (assume (nonzero (cos (tanh x))))
  (ask (= (atanh (sin (tanh x))) (asinh (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.tanh r := (by linarith [Real.neg_one_lt_tanh r, Real.pi_gt_three]); have hy2 : Real.tanh r < Real.pi / 2 := (by linarith [Real.tanh_lt_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.tanh_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.tanh r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.tanh r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp [AE.cosh_ofReal_ne_zero]⟩, ?_⟩; rw [AE.tanh_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-218-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-6}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-6}{x}} = \frac{x}{-6}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0 -> / !0 (-6)

Lean proof: included

(case sp-acj4-218-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0  ->  / !0 (-6)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -6 x)))
  (ask (= (/ (/ -6 x)) (/ x -6)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-22-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \tanh\left(\left|x\right|\right) = \left|\tanh x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0

Lean proof: included

(case sp-acj4-22-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0  ->  abs tanh _0")
  (var x real)
  (ask (= (tanh (abs x)) (abs (tanh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [AE.abs_ofReal, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal]; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.tanh_abs]"))

sp-acj4-220-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-8}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-8}{x}} = \frac{x}{-8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0 -> / !0 (-8)

Lean proof: included

(case sp-acj4-220-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0  ->  / !0 (-8)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -8 x)))
  (ask (= (/ (/ -8 x)) (/ x -8)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-233-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(-\pi\right) > 0\)

Claim

\(\displaystyle \frac{1}{x^{\pi}} = x^{-\pi}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi -> pow _0 neg np.pi

Lean proof: included

(case sp-acj4-233-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi  ->  pow _0 neg np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (assume (nonzero (pow x pi)))
  (assume (or (nonzero x) (> (re (- pi)) 0)))
  (ask (= (/ (pow x pi)) (pow x (- pi))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := (by intro h; apply h1.2; rw [h]; simp); refine ⟨⟨⟨by simp [hx], h1.2⟩, by simp [hx]⟩, ?_⟩; simp only [AE.pow, if_neg hx, AE.inv_def, AE.exp_def]; rw [← Complex.exp_neg]; ring_nf"))

sp-acj4-248-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\arcsin x\right| \overset{?}{=} \arcsin\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0 -> asin neg abs _0

Lean proof: included

(case sp-acj4-248-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0  ->  asin neg abs _0")
  (var x real)
  (ask (= (- (abs (asin x))) (asin (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r248w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-249-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\operatorname{arsinh}\left(x\right)\right| = \operatorname{arsinh}\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0 -> asinh neg abs _0

Lean proof: included

(case sp-acj4-249-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0  ->  asinh neg abs _0")
  (var x real)
  (ask (= (- (abs (asinh x))) (asinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.asinh_ofReal, AE.asinh_ofReal, AE.abs_ofReal, Real.arsinh_neg, AE.arsinh_abs]; push_cast; ring"))

sp-acj4-251-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1,\quad -\left|x\right| \neq 1,\quad -\left|x\right| \neq -1\)

Claim

\(\displaystyle -\left|\operatorname{artanh}\left(x\right)\right| \overset{?}{=} \operatorname{artanh}\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0 -> atanh neg abs _0

Lean proof: included

(case sp-acj4-251-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0  ->  atanh neg abs _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (assume (!= (- (abs x)) 1))
  (assume (!= (- (abs x)) -1))
  (ask (= (- (abs (atanh x))) (atanh (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r251w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-252-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\sinh x\right| = \sinh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0

Lean proof: included

(case sp-acj4-252-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0  ->  sinh neg abs _0")
  (var x real)
  (ask (= (- (abs (sinh x))) (sinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.sinh_ofReal, AE.sinh_ofReal, AE.abs_ofReal, Real.abs_sinh, Real.sinh_neg]; push_cast; ring"))

sp-acj4-253-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\tanh x\right| = \tanh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0

Lean proof: included

(case sp-acj4-253-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0  ->  tanh neg abs _0")
  (var x real)
  (ask (= (- (abs (tanh x))) (tanh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal, ← AE.tanh_abs, Real.tanh_neg]; push_cast; ring"))

sp-acj4-256-real

Given

\(x \in \mathbb{R},\quad -e^{x} \neq 0\)

Claim

\(\displaystyle -e^{-x} = \frac{1}{-e^{x}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0 -> inv neg exp !0

Lean proof: included

(case sp-acj4-256-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0  ->  inv neg exp !0")
  (var x real)
  (assume (nonzero (- (exp x))))
  (ask (= (- (exp (- x))) (/ (- (exp x)))))
  (expect true)
  (lean-proof "refine ⟨h0.2, ?_⟩; simp [Complex.exp_neg]"))

sp-acj4-275-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-3} = \left|x^{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3) -> abs pow _0 (-3)

Lean proof: included

(case sp-acj4-275-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3)  ->  abs pow _0 (-3)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -3) (abs (pow x -3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr], by simp⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_inv, AE.abs_eq, AE.abs_eq]; simp"))

sp-acj4-276-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-4} = x^{-4}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4) -> pow _0 (-4)

Lean proof: included

(case sp-acj4-276-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4)  ->  pow _0 (-4)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -4) (pow x -4)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr]⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_ofReal]; congr 1; exact_mod_cast (Even.pow_abs ⟨2, by norm_num⟩ r)"))

sp-acj4-283-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{2} = x \, x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0

Lean proof: included

(case sp-acj4-283-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2  ->  * _0 _0")
  (var x real)
  (ask (= (pow (abs x) 2) (* x x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (by rw [sq_abs]; ring : |r|^2 = r*r)"))

sp-acj4-286-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{5} = \left|x^{5}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5 -> abs pow _0 5

Lean proof: included

(case sp-acj4-286-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5  ->  abs pow _0 5")
  (var x real)
  (ask (= (pow (abs x) 5) (abs (pow x 5))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal r, ← Complex.ofReal_pow r, AE.abs_ofReal (r^5), abs_pow]; push_cast; ring"))

sp-acj4-289-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{8} = x^{8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8 -> pow _0 8

Lean proof: included

(case sp-acj4-289-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8  ->  pow _0 8")
  (var x real)
  (ask (= (pow (abs x) 8) (pow x 8)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (Even.pow_abs ⟨4, by norm_num⟩ r)"))

sp-acj4-31-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 5 \, \left|x\right| = \left|-5 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0 -> abs * (-5) _0

Lean proof: included

(case sp-acj4-31-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0  ->  abs * (-5) _0")
  (var x real)
  (ask (= (* 5 (abs x)) (abs (* -5 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-326-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left(e^{x}\right)^{2} = e^{2 \, x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2 -> exp * 2 _0

Lean proof: included

(case sp-acj4-326-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2  ->  exp * 2 _0")
  (var x real)
  (ask (= (pow (exp x) 2) (exp (* 2 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.exp_def, AE.exp_def, ← Complex.exp_nat_mul]; norm_num"))

sp-acj4-33-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 6 \, \left|x\right| = \left|-6 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0 -> abs * (-6) _0

Lean proof: included

(case sp-acj4-33-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0  ->  abs * (-6) _0")
  (var x real)
  (ask (= (* 6 (abs x)) (abs (* -6 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-35-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{7} \neq 0\)

Claim

\(\displaystyle 7 \, \log x \overset{?}{=} \log\left(x^{7}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0 -> log pow _0 7

Lean proof: included

(case sp-acj4-35-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0  ->  log pow _0 7")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 7)))
  (ask (= (* 7 (log x)) (log (pow x 7))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r35w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-354-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 5\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 5} = 5^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5 -> pow 5 log _0

Lean proof: included

(case sp-acj4-354-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5  ->  pow 5 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 5)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 5)) (pow 5 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-356-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 7\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 7} = 7^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7 -> pow 7 log _0

Lean proof: included

(case sp-acj4-356-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7  ->  pow 7 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 7)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 7)) (pow 7 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-359-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log \pi\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log \pi} = \pi^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi -> pow np.pi log _0

Lean proof: included

(case sp-acj4-359-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi  ->  pow np.pi log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log pi)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log pi)) (pow pi (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by simp [Real.pi_ne_zero], by simp [hx]⟩, hx, by simp [Real.pi_ne_zero]⟩, ?_⟩; simp [AE.pow, hx, Real.pi_ne_zero, mul_comm]"))

sp-acj4-364-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 3\right) = \left|\operatorname{surd}\left(x, 3\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3 -> abs rootn _0 3

Lean proof: included

(case sp-acj4-364-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3  ->  abs rootn _0 3")
  (var x real)
  (ask (= (surd (abs x) 3) (abs (surd x 3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (3:ℂ) = ((3:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-367-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 9\right) = \left|\operatorname{surd}\left(x, 9\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9 -> abs rootn _0 9

Lean proof: included

(case sp-acj4-367-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9  ->  abs rootn _0 9")
  (var x real)
  (ask (= (surd (abs x) 9) (abs (surd x 9))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (9:ℂ) = ((9:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-368-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2

Lean proof: included

(case sp-acj4-368-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5)  ->  pow acos _0 -0.2")
  (var x real)
  (assume (real (acos x)))
  (assume (nonzero (acos x)))
  (ask (= (surd (acos x) -5) (pow (acos x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; have hne : Real.arccos r ≠ 0 := (by have := h1.2; rw [AE.acos_ofReal hI.1 hI.2] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-369-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2

Lean proof: included

(case sp-acj4-369-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5  ->  pow acos _0 0.2")
  (var x real)
  (assume (real (acos x)))
  (ask (= (surd (acos x) 5) (pow (acos x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-370-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2

Lean proof: included

(case sp-acj4-370-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5)  ->  pow acosh _0 -0.2")
  (var x real)
  (assume (real (acosh x)))
  (assume (nonzero (acosh x)))
  (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; have hne : Real.arcosh r ≠ 0 := (by have := h1.2; rw [AE.acosh_ofReal hI] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-371-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2

Lean proof: included

(case sp-acj4-371-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5  ->  pow acosh _0 0.2")
  (var x real)
  (assume (real (acosh x)))
  (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-372-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) = \left(\cosh x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2

Lean proof: included

(case sp-acj4-372-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5)  ->  pow cosh _0 -0.2")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-373-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) = \left(\cosh x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2

Lean proof: included

(case sp-acj4-373-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5  ->  pow cosh _0 0.2")
  (var x real)
  (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-38-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{9} \neq 0\)

Claim

\(\displaystyle 9 \, \log x \overset{?}{=} \log\left(x^{9}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0 -> log pow _0 9

Lean proof: included

(case sp-acj4-38-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0  ->  log pow _0 9")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 9)))
  (ask (= (* 9 (log x)) (log (pow x 9))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r38w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-394-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\left|\operatorname{artanh}\left(x\right)\right|\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-394-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (abs (atanh x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r394w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-395-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\cos x\right|\right) = \left|\sin\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0 -> abs sin cos _0

Lean proof: included

(case sp-acj4-395-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0  ->  abs sin cos _0")
  (var x real)
  (ask (= (sin (abs (cos x))) (abs (sin (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.cos_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-396-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\sin x\right|\right) = \left|\sin\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0

Lean proof: included

(case sp-acj4-396-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0  ->  abs sin sin _0")
  (var x real)
  (ask (= (sin (abs (sin x))) (abs (sin (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.sin_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-397-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\tanh x\right|\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-397-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (abs (tanh x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-402-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\operatorname{artanh}\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-402-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (atanh (abs x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r402w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-404-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-404-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (tanh (abs x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.tanh_abs, AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-412-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\arcsin x\right|\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\arcsin x\right|\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-412-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (abs (asin x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (abs (asin x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r412w2 | exact AE.w2_r412_cos | exact AE.w2_r412_tanD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-413-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\cos x\right|\right) \neq 0,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\cos x\right|\right) = \left|\tan\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0 -> abs tan cos _0

Lean proof: included

(case sp-acj4-413-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0  ->  abs tan cos _0")
  (var x real)
  (assume (nonzero (cos (abs (cos x)))))
  (assume (nonzero (cos (cos x))))
  (ask (= (tan (abs (cos x))) (abs (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.cos_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-414-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\sin x\right|\right) = \left|\tan\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0

Lean proof: included

(case sp-acj4-414-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0  ->  abs tan sin _0")
  (var x real)
  (assume (nonzero (cos (abs (sin x)))))
  (assume (nonzero (cos (sin x))))
  (ask (= (tan (abs (sin x))) (abs (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.sin_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-415-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\tanh x\right|\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-415-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (abs (tanh x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (abs (tanh x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.tanh_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-417-real

Given

\(x \in \mathbb{R},\quad \cos\left(\arcsin\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\arcsin\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-417-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (asin (abs x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (asin (abs x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r417w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-421-real

Given

\(x \in \mathbb{R},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-421-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (tanh (abs x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (tanh (abs x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.abs_ofReal, AE.tanh_ofReal, AE.tan_ofReal]; rw [AE.tanh_abs, AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-5-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \operatorname{artanh}\left(\left|x\right|\right) \overset{?}{=} \left|\operatorname{artanh}\left(x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0 -> abs atanh _0

Lean proof: included

(case sp-acj4-5-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0  ->  abs atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (atanh (abs x)) (abs (atanh x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r5w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-73-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad -\left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{-1}{\left|x\right|} = \frac{1}{-\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0 -> inv neg abs !0

Lean proof: included

(case sp-acj4-73-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0  ->  inv neg abs !0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero (- (abs x))))
  (ask (= (/ -1 (abs x)) (/ (- (abs x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-80-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad -\cos x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\cos x} = \frac{1}{-\cos x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0 -> inv neg cos _0

Lean proof: included

(case sp-acj4-80-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0  ->  inv neg cos _0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (- (cos x))))
  (ask (= (/ -1 (cos x)) (/ (- (cos x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-83-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \log x \neq 0,\quad -\log x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\log x} = \frac{1}{-\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0 -> inv neg log !0

Lean proof: included

(case sp-acj4-83-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0  ->  inv neg log !0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (log x)))
  (assume (nonzero (- (log x))))
  (ask (= (/ -1 (log x)) (/ (- (log x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h0.2, h1.2⟩, h0.2, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-86-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad \tan x \neq 0,\quad -\tan x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\tan x} = \frac{1}{-\tan x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0 -> inv neg tan !0

Lean proof: included

(case sp-acj4-86-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0  ->  inv neg tan !0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (tan x)))
  (assume (nonzero (- (tan x))))
  (ask (= (/ -1 (tan x)) (/ (- (tan x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h1.1.1, h1.2⟩, h1.1.1, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-90-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0

Lean proof: included

(case sp-acj4-90-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0  ->  abs / (-3) _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ 3 (abs x)) (abs (/ -3 x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-99-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)

Lean proof: included

(case sp-acj4-99-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3  ->  abs / _0 (-3)")
  (var x real)
  (ask (= (/ (abs x) 3) (abs (/ x -3))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

Case set `3ba0dfac4bca`

case	sympy best	sympy-ask	sympy-refine	sympy-simplify	wolfram best	wolfram-fullsimplify	wolfram-refine	wolfram-simplify
sp-acj4-1-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-102-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-108-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	no verdict	true ✓
sp-acj4-120-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-127-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-128-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-131-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-132-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-133-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-136-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-137-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-14-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-145-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-162-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-179-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-181-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-185-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-194-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-195-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-196-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-218-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-22-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-220-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-233-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-248-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-249-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-251-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-252-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-253-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-256-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-275-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-276-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-283-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-286-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-289-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-31-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-326-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-33-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-35-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-354-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-356-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-359-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-364-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-367-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-368-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-369-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-370-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-371-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-372-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-373-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-38-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-394-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-395-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-396-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-397-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-402-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-404-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-412-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-413-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-414-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-415-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-417-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-421-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-5-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-73-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-80-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-83-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-86-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-90-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-99-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓

sp-acj4-1-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \arcsin\left(\left|x\right|\right) \overset{?}{=} \left|\arcsin x\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0 -> abs asin _0

Lean proof: included

(case sp-acj4-1-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0  ->  abs asin _0")
  (var x real)
  (ask (= (asin (abs x)) (abs (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r1w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-102-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)

Lean proof: included

(case sp-acj4-102-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6  ->  abs / _0 (-6)")
  (var x real)
  (ask (= (/ (abs x) 6) (abs (/ x -6))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-108-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{\left|x\right|}{x} = \frac{x}{\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0

Lean proof: included

(case sp-acj4-108-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0  ->  / _0 abs _0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (abs x)))
  (ask (= (/ (abs x) x) (/ x (abs x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; have hr : (r:ℂ) ≠ 0 := h0.2; have ha : ((|r|:ℝ):ℂ) ≠ 0 := (by have := h1.2; rwa [AE.abs_ofReal] at this); simp only [AE.abs_ofReal, AE.inv_def]; rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_eq_div_iff hr ha]; exact_mod_cast (abs_mul_abs_self r)"))

sp-acj4-120-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0

Lean proof: included

(case sp-acj4-120-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0  ->  abs / np.pi _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ pi (abs x)) (abs (/ pi x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, abs_of_pos Real.pi_pos]"))

sp-acj4-127-real

Given

\(x \in \mathbb{R},\quad e^{x} \neq 1,\quad e^{x} \neq -1\)

Claim

\(\displaystyle \left|\operatorname{artanh}\left(e^{x}\right)\right| \overset{?}{=} \operatorname{artanh}\left(e^{x}\right)\)

Expected

contingent true at \(x = -1\); false at \(x = 1\)

equal at some admissible points, different at others

witness (true): \(x = -1\)

witness (false): \(x = 1\)

SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0 -> atanh exp _0

Lean proof: included

(case sp-acj4-127-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0  ->  atanh exp _0")
  (var x real)
  (assume (!= (exp x) 1))
  (assume (!= (exp x) -1))
  (ask (= (abs (atanh (exp x))) (atanh (exp x))))
  (expect contingent)
  (witness true (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w1 | exact AE.w2_exp_neg_one_ne_one | exact AE.w2_exp_neg_one_ne_neg_one | exact AE.w2_atanhD_exp_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w2 | exact AE.w2_exp_one_ne_one | exact AE.w2_exp_one_ne_neg_one | exact AE.w2_atanhD_exp_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-128-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\cos\left(\arcsin x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0 -> cos asin _0

Lean proof: included

(case sp-acj4-128-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (cos (asin x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r128w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-131-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\cosh x\right)\right| = \log\left(\cosh x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0

Lean proof: included

(case sp-acj4-131-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0  ->  log cosh _0")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (abs (log (cosh x))) (log (cosh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h0.2⟩, by simp⟩, trivial, h0.2⟩, ?_⟩; rw [AE.cosh_ofReal, AE.log_ofReal (Real.cosh_pos r).le, AE.abs_ofReal, abs_of_nonneg (Real.log_nonneg (Real.one_le_cosh r))]"))

sp-acj4-132-real

Given

\(x \in \mathbb{R},\quad \sin x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0

Lean proof: included

(case sp-acj4-132-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0  ->  neg log sin _0")
  (var x real)
  (assume (nonzero (sin x)))
  (ask (= (abs (log (sin x))) (- (log (sin x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w1 | exact AE.w2_sin_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w2 | exact AE.w2_sin_neg_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-133-real

Given

\(x \in \mathbb{R},\quad \tanh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0

Lean proof: included

(case sp-acj4-133-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0  ->  neg log tanh _0")
  (var x real)
  (assume (nonzero (tanh x)))
  (ask (= (abs (log (tanh x))) (- (log (tanh x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w1 | exact AE.w2_tanh_one_ne | exact AE.w2_tanhD_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w2 | exact AE.w2_tanh_neg_one_ne | exact AE.w2_tanhD_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-136-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)

Claim

\(\displaystyle \left|x^{e^{1}}\right| \overset{?}{=} x^{e^{1}}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e -> pow _0 np.e

Lean proof: included

(case sp-acj4-136-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e  ->  pow _0 np.e")
  (var x real)
  (assume (or (nonzero x) (> (re (exp 1)) 0)))
  (ask (= (abs (pow x (exp 1))) (pow x (exp 1))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first  | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, Complex.exp_im]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r136w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-137-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)

Claim

\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi

Lean proof: included

(case sp-acj4-137-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi  ->  pow _0 np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (ask (= (abs (pow x pi)) (pow x pi)))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r137w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-14-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\sin x = \sin\left(-x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0 -> sin neg _0

Lean proof: included

(case sp-acj4-14-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0  ->  sin neg _0")
  (var x real)
  (ask (= (- (sin x)) (sin (- x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [Complex.sin_neg]"))

sp-acj4-145-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\sin\left(\arccos x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0 -> cos asin _0

Lean proof: included

(case sp-acj4-145-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (sin (acos x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r145w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-162-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{arcosh}\left(\operatorname{surd}\left(x, 5\right)\right) \overset{?}{=} \operatorname{arcosh}\left(x^{\frac{1}{5}}\right)\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5 -> acosh pow _0 0.2

Lean proof: included

(case sp-acj4-162-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5  ->  acosh pow _0 0.2")
  (var x real)
  (ask (= (acosh (surd x 5)) (acosh (pow x 1/5))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r162w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-179-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arccos x\right) \neq i,\quad \sinh\left(\arccos x\right) \neq -i,\quad \cosh\left(\arccos x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arccos x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arccos x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0 -> asin tanh acos _0

Lean proof: included

(case sp-acj4-179-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0  ->  asin tanh acos _0")
  (var x real)
  (assume (!= (sinh (acos x)) i))
  (assume (!= (sinh (acos x)) (- i)))
  (assume (nonzero (cosh (acos x))))
  (ask (= (atan (sinh (acos x))) (asin (tanh (acos x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w1 | exact AE.w2_r179w1_hI | exact AE.w2_r179w1_hnI | exact AE.w2_r179w1_atanD | exact AE.w2_r179w1_cosh | exact AE.w2_r179w1_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w2 | exact AE.w2_r179_hI | exact AE.w2_r179_hnI | exact AE.w2_r179_atanD | exact AE.w2_r179_cosh | exact AE.w2_r179_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-181-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arcsin x\right) \neq i,\quad \sinh\left(\arcsin x\right) \neq -i,\quad \cosh\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arcsin x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arcsin x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0 -> asin tanh asin _0

Lean proof: included

(case sp-acj4-181-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0  ->  asin tanh asin _0")
  (var x real)
  (assume (!= (sinh (asin x)) i))
  (assume (!= (sinh (asin x)) (- i)))
  (assume (nonzero (cosh (asin x))))
  (ask (= (atan (sinh (asin x))) (asin (tanh (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r181w2 | exact AE.w2_r181_hI | exact AE.w2_r181_hnI | exact AE.w2_r181_atanD | exact AE.w2_r181_cosh | exact AE.w2_r181_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-185-real

Given

\(x \in \mathbb{R},\quad \cos\left(\cos x\right) \neq 0,\quad \tan\left(\cos x\right) \neq i,\quad \tan\left(\cos x\right) \neq -i\)

Claim

\(\displaystyle \arctan\left(\tan\left(\cos x\right)\right) = \cos x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0 -> cos _0

Lean proof: included

(case sp-acj4-185-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0  ->  cos _0")
  (var x real)
  (assume (nonzero (cos (cos x))))
  (assume (!= (tan (cos x)) i))
  (assume (!= (tan (cos x)) (- i)))
  (ask (= (atan (tan (cos x))) (cos x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h1.1.2⟩, by rw [AE.cos_ofReal, AE.tan_ofReal]; exact AE.atan.D_ofReal _⟩, trivial⟩, ?_⟩; rw [AE.cos_ofReal, AE.tan_ofReal, AE.atan_ofReal, Real.arctan_tan (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]) (by linarith [Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-194-real

Given

\(x \in \mathbb{R},\quad \sin\left(\cos x\right) \neq 1,\quad \sin\left(\cos x\right) \neq -1,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\cos x\right)\right) = \operatorname{arsinh}\left(\tan\left(\cos x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0 -> asinh tan cos _0

Lean proof: included

(case sp-acj4-194-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0  ->  asinh tan cos _0")
  (var x real)
  (assume (!= (sin (cos x)) 1))
  (assume (!= (sin (cos x)) -1))
  (assume (nonzero (cos (cos x))))
  (ask (= (atanh (sin (cos x))) (asinh (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.cos r := (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]); have hy2 : Real.cos r < Real.pi / 2 := (by linarith [Real.cos_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.cos_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.cos r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.cos r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.cos_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-195-real

Given

\(x \in \mathbb{R},\quad \sin\left(\sin x\right) \neq 1,\quad \sin\left(\sin x\right) \neq -1,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\sin x\right)\right) = \operatorname{arsinh}\left(\tan\left(\sin x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0 -> asinh tan sin _0

Lean proof: included

(case sp-acj4-195-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0  ->  asinh tan sin _0")
  (var x real)
  (assume (!= (sin (sin x)) 1))
  (assume (!= (sin (sin x)) -1))
  (assume (nonzero (cos (sin x))))
  (ask (= (atanh (sin (sin x))) (asinh (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.sin r := (by linarith [Real.neg_one_le_sin r, Real.pi_gt_three]); have hy2 : Real.sin r < Real.pi / 2 := (by linarith [Real.sin_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.sin_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.sin r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.sin r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.sin_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-196-real

Given

\(x \in \mathbb{R},\quad \sin\left(\tanh x\right) \neq 1,\quad \sin\left(\tanh x\right) \neq -1,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\tanh x\right)\right) = \operatorname{arsinh}\left(\tan\left(\tanh x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0 -> asinh tan tanh _0

Lean proof: included

(case sp-acj4-196-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0  ->  asinh tan tanh _0")
  (var x real)
  (assume (!= (sin (tanh x)) 1))
  (assume (!= (sin (tanh x)) -1))
  (assume (nonzero (cos (tanh x))))
  (ask (= (atanh (sin (tanh x))) (asinh (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.tanh r := (by linarith [Real.neg_one_lt_tanh r, Real.pi_gt_three]); have hy2 : Real.tanh r < Real.pi / 2 := (by linarith [Real.tanh_lt_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.tanh_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.tanh r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.tanh r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp [AE.cosh_ofReal_ne_zero]⟩, ?_⟩; rw [AE.tanh_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-218-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-6}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-6}{x}} = \frac{x}{-6}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0 -> / !0 (-6)

Lean proof: included

(case sp-acj4-218-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0  ->  / !0 (-6)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -6 x)))
  (ask (= (/ (/ -6 x)) (/ x -6)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-22-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \tanh\left(\left|x\right|\right) = \left|\tanh x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0

Lean proof: included

(case sp-acj4-22-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0  ->  abs tanh _0")
  (var x real)
  (ask (= (tanh (abs x)) (abs (tanh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [AE.abs_ofReal, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal]; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.tanh_abs]"))

sp-acj4-220-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-8}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-8}{x}} = \frac{x}{-8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0 -> / !0 (-8)

Lean proof: included

(case sp-acj4-220-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0  ->  / !0 (-8)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -8 x)))
  (ask (= (/ (/ -8 x)) (/ x -8)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-233-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(-\pi\right) > 0\)

Claim

\(\displaystyle \frac{1}{x^{\pi}} = x^{-\pi}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi -> pow _0 neg np.pi

Lean proof: included

(case sp-acj4-233-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi  ->  pow _0 neg np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (assume (nonzero (pow x pi)))
  (assume (or (nonzero x) (> (re (- pi)) 0)))
  (ask (= (/ (pow x pi)) (pow x (- pi))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := (by intro h; apply h1.2; rw [h]; simp); refine ⟨⟨⟨by simp [hx], h1.2⟩, by simp [hx]⟩, ?_⟩; simp only [AE.pow, if_neg hx, AE.inv_def, AE.exp_def]; rw [← Complex.exp_neg]; ring_nf"))

sp-acj4-248-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\arcsin x\right| \overset{?}{=} \arcsin\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0 -> asin neg abs _0

Lean proof: included

(case sp-acj4-248-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0  ->  asin neg abs _0")
  (var x real)
  (ask (= (- (abs (asin x))) (asin (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r248w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-249-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\operatorname{arsinh}\left(x\right)\right| = \operatorname{arsinh}\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0 -> asinh neg abs _0

Lean proof: included

(case sp-acj4-249-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0  ->  asinh neg abs _0")
  (var x real)
  (ask (= (- (abs (asinh x))) (asinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.asinh_ofReal, AE.asinh_ofReal, AE.abs_ofReal, Real.arsinh_neg, AE.arsinh_abs]; push_cast; ring"))

sp-acj4-251-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1,\quad -\left|x\right| \neq 1,\quad -\left|x\right| \neq -1\)

Claim

\(\displaystyle -\left|\operatorname{artanh}\left(x\right)\right| \overset{?}{=} \operatorname{artanh}\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0 -> atanh neg abs _0

Lean proof: included

(case sp-acj4-251-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0  ->  atanh neg abs _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (assume (!= (- (abs x)) 1))
  (assume (!= (- (abs x)) -1))
  (ask (= (- (abs (atanh x))) (atanh (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r251w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-252-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\sinh x\right| = \sinh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0

Lean proof: included

(case sp-acj4-252-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0  ->  sinh neg abs _0")
  (var x real)
  (ask (= (- (abs (sinh x))) (sinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.sinh_ofReal, AE.sinh_ofReal, AE.abs_ofReal, Real.abs_sinh, Real.sinh_neg]; push_cast; ring"))

sp-acj4-253-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\tanh x\right| = \tanh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0

Lean proof: included

(case sp-acj4-253-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0  ->  tanh neg abs _0")
  (var x real)
  (ask (= (- (abs (tanh x))) (tanh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal, ← AE.tanh_abs, Real.tanh_neg]; push_cast; ring"))

sp-acj4-256-real

Given

\(x \in \mathbb{R},\quad -e^{x} \neq 0\)

Claim

\(\displaystyle -e^{-x} = \frac{1}{-e^{x}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0 -> inv neg exp !0

Lean proof: included

(case sp-acj4-256-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0  ->  inv neg exp !0")
  (var x real)
  (assume (nonzero (- (exp x))))
  (ask (= (- (exp (- x))) (/ (- (exp x)))))
  (expect true)
  (lean-proof "refine ⟨h0.2, ?_⟩; simp [Complex.exp_neg]"))

sp-acj4-275-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-3} = \left|x^{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3) -> abs pow _0 (-3)

Lean proof: included

(case sp-acj4-275-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3)  ->  abs pow _0 (-3)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -3) (abs (pow x -3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr], by simp⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_inv, AE.abs_eq, AE.abs_eq]; simp"))

sp-acj4-276-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-4} = x^{-4}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4) -> pow _0 (-4)

Lean proof: included

(case sp-acj4-276-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4)  ->  pow _0 (-4)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -4) (pow x -4)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr]⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_ofReal]; congr 1; exact_mod_cast (Even.pow_abs ⟨2, by norm_num⟩ r)"))

sp-acj4-283-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{2} = x \, x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0

Lean proof: included

(case sp-acj4-283-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2  ->  * _0 _0")
  (var x real)
  (ask (= (pow (abs x) 2) (* x x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (by rw [sq_abs]; ring : |r|^2 = r*r)"))

sp-acj4-286-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{5} = \left|x^{5}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5 -> abs pow _0 5

Lean proof: included

(case sp-acj4-286-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5  ->  abs pow _0 5")
  (var x real)
  (ask (= (pow (abs x) 5) (abs (pow x 5))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal r, ← Complex.ofReal_pow r, AE.abs_ofReal (r^5), abs_pow]; push_cast; ring"))

sp-acj4-289-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{8} = x^{8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8 -> pow _0 8

Lean proof: included

(case sp-acj4-289-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8  ->  pow _0 8")
  (var x real)
  (ask (= (pow (abs x) 8) (pow x 8)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (Even.pow_abs ⟨4, by norm_num⟩ r)"))

sp-acj4-31-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 5 \, \left|x\right| = \left|-5 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0 -> abs * (-5) _0

Lean proof: included

(case sp-acj4-31-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0  ->  abs * (-5) _0")
  (var x real)
  (ask (= (* 5 (abs x)) (abs (* -5 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-326-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left(e^{x}\right)^{2} = e^{2 \, x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2 -> exp * 2 _0

Lean proof: included

(case sp-acj4-326-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2  ->  exp * 2 _0")
  (var x real)
  (ask (= (pow (exp x) 2) (exp (* 2 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.exp_def, AE.exp_def, ← Complex.exp_nat_mul]; norm_num"))

sp-acj4-33-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 6 \, \left|x\right| = \left|-6 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0 -> abs * (-6) _0

Lean proof: included

(case sp-acj4-33-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0  ->  abs * (-6) _0")
  (var x real)
  (ask (= (* 6 (abs x)) (abs (* -6 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-35-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{7} \neq 0\)

Claim

\(\displaystyle 7 \, \log x \overset{?}{=} \log\left(x^{7}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0 -> log pow _0 7

Lean proof: included

(case sp-acj4-35-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0  ->  log pow _0 7")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 7)))
  (ask (= (* 7 (log x)) (log (pow x 7))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r35w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-354-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 5\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 5} = 5^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5 -> pow 5 log _0

Lean proof: included

(case sp-acj4-354-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5  ->  pow 5 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 5)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 5)) (pow 5 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-356-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 7\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 7} = 7^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7 -> pow 7 log _0

Lean proof: included

(case sp-acj4-356-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7  ->  pow 7 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 7)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 7)) (pow 7 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-359-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log \pi\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log \pi} = \pi^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi -> pow np.pi log _0

Lean proof: included

(case sp-acj4-359-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi  ->  pow np.pi log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log pi)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log pi)) (pow pi (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by simp [Real.pi_ne_zero], by simp [hx]⟩, hx, by simp [Real.pi_ne_zero]⟩, ?_⟩; simp [AE.pow, hx, Real.pi_ne_zero, mul_comm]"))

sp-acj4-364-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 3\right) = \left|\operatorname{surd}\left(x, 3\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3 -> abs rootn _0 3

Lean proof: included

(case sp-acj4-364-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3  ->  abs rootn _0 3")
  (var x real)
  (ask (= (surd (abs x) 3) (abs (surd x 3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (3:ℂ) = ((3:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-367-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 9\right) = \left|\operatorname{surd}\left(x, 9\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9 -> abs rootn _0 9

Lean proof: included

(case sp-acj4-367-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9  ->  abs rootn _0 9")
  (var x real)
  (ask (= (surd (abs x) 9) (abs (surd x 9))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (9:ℂ) = ((9:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-368-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2

Lean proof: included

(case sp-acj4-368-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5)  ->  pow acos _0 -0.2")
  (var x real)
  (assume (real (acos x)))
  (assume (nonzero (acos x)))
  (ask (= (surd (acos x) -5) (pow (acos x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; have hne : Real.arccos r ≠ 0 := (by have := h1.2; rw [AE.acos_ofReal hI.1 hI.2] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-369-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2

Lean proof: included

(case sp-acj4-369-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5  ->  pow acos _0 0.2")
  (var x real)
  (assume (real (acos x)))
  (ask (= (surd (acos x) 5) (pow (acos x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-370-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2

Lean proof: included

(case sp-acj4-370-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5)  ->  pow acosh _0 -0.2")
  (var x real)
  (assume (real (acosh x)))
  (assume (nonzero (acosh x)))
  (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; have hne : Real.arcosh r ≠ 0 := (by have := h1.2; rw [AE.acosh_ofReal hI] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-371-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2

Lean proof: included

(case sp-acj4-371-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5  ->  pow acosh _0 0.2")
  (var x real)
  (assume (real (acosh x)))
  (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-372-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) = \left(\cosh x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2

Lean proof: included

(case sp-acj4-372-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5)  ->  pow cosh _0 -0.2")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-373-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) = \left(\cosh x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2

Lean proof: included

(case sp-acj4-373-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5  ->  pow cosh _0 0.2")
  (var x real)
  (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-38-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{9} \neq 0\)

Claim

\(\displaystyle 9 \, \log x \overset{?}{=} \log\left(x^{9}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0 -> log pow _0 9

Lean proof: included

(case sp-acj4-38-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0  ->  log pow _0 9")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 9)))
  (ask (= (* 9 (log x)) (log (pow x 9))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r38w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-394-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\left|\operatorname{artanh}\left(x\right)\right|\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-394-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (abs (atanh x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r394w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-395-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\cos x\right|\right) = \left|\sin\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0 -> abs sin cos _0

Lean proof: included

(case sp-acj4-395-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0  ->  abs sin cos _0")
  (var x real)
  (ask (= (sin (abs (cos x))) (abs (sin (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.cos_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-396-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\sin x\right|\right) = \left|\sin\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0

Lean proof: included

(case sp-acj4-396-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0  ->  abs sin sin _0")
  (var x real)
  (ask (= (sin (abs (sin x))) (abs (sin (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.sin_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-397-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\tanh x\right|\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-397-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (abs (tanh x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-402-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\operatorname{artanh}\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-402-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (atanh (abs x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r402w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-404-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-404-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (tanh (abs x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.tanh_abs, AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-412-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\arcsin x\right|\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\arcsin x\right|\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-412-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (abs (asin x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (abs (asin x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r412w2 | exact AE.w2_r412_cos | exact AE.w2_r412_tanD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-413-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\cos x\right|\right) \neq 0,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\cos x\right|\right) = \left|\tan\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0 -> abs tan cos _0

Lean proof: included

(case sp-acj4-413-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0  ->  abs tan cos _0")
  (var x real)
  (assume (nonzero (cos (abs (cos x)))))
  (assume (nonzero (cos (cos x))))
  (ask (= (tan (abs (cos x))) (abs (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.cos_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-414-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\sin x\right|\right) = \left|\tan\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0

Lean proof: included

(case sp-acj4-414-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0  ->  abs tan sin _0")
  (var x real)
  (assume (nonzero (cos (abs (sin x)))))
  (assume (nonzero (cos (sin x))))
  (ask (= (tan (abs (sin x))) (abs (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.sin_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-415-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\tanh x\right|\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-415-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (abs (tanh x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (abs (tanh x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.tanh_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-417-real

Given

\(x \in \mathbb{R},\quad \cos\left(\arcsin\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\arcsin\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-417-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (asin (abs x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (asin (abs x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r417w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-421-real

Given

\(x \in \mathbb{R},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-421-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (tanh (abs x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (tanh (abs x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.abs_ofReal, AE.tanh_ofReal, AE.tan_ofReal]; rw [AE.tanh_abs, AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-5-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \operatorname{artanh}\left(\left|x\right|\right) \overset{?}{=} \left|\operatorname{artanh}\left(x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0 -> abs atanh _0

Lean proof: included

(case sp-acj4-5-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0  ->  abs atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (atanh (abs x)) (abs (atanh x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r5w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-73-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad -\left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{-1}{\left|x\right|} = \frac{1}{-\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0 -> inv neg abs !0

Lean proof: included

(case sp-acj4-73-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0  ->  inv neg abs !0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero (- (abs x))))
  (ask (= (/ -1 (abs x)) (/ (- (abs x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-80-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad -\cos x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\cos x} = \frac{1}{-\cos x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0 -> inv neg cos _0

Lean proof: included

(case sp-acj4-80-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0  ->  inv neg cos _0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (- (cos x))))
  (ask (= (/ -1 (cos x)) (/ (- (cos x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-83-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \log x \neq 0,\quad -\log x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\log x} = \frac{1}{-\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0 -> inv neg log !0

Lean proof: included

(case sp-acj4-83-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0  ->  inv neg log !0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (log x)))
  (assume (nonzero (- (log x))))
  (ask (= (/ -1 (log x)) (/ (- (log x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h0.2, h1.2⟩, h0.2, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-86-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad \tan x \neq 0,\quad -\tan x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\tan x} = \frac{1}{-\tan x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0 -> inv neg tan !0

Lean proof: included

(case sp-acj4-86-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0  ->  inv neg tan !0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (tan x)))
  (assume (nonzero (- (tan x))))
  (ask (= (/ -1 (tan x)) (/ (- (tan x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h1.1.1, h1.2⟩, h1.1.1, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-90-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0

Lean proof: included

(case sp-acj4-90-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0  ->  abs / (-3) _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ 3 (abs x)) (abs (/ -3 x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-99-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)

Lean proof: included

(case sp-acj4-99-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3  ->  abs / _0 (-3)")
  (var x real)
  (ask (= (/ (abs x) 3) (abs (/ x -3))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

Case set `ee702ae2fe06`

case	sympy best	sympy-ask	sympy-refine	sympy-simplify	wolfram best	wolfram-fullsimplify	wolfram-refine	wolfram-simplify
sp-acj4-1-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-102-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-108-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	no verdict	true ✓
sp-acj4-120-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-127-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-128-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-131-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-132-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-133-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-136-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-137-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-14-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-145-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-162-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-179-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-181-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-185-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓
sp-acj4-194-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-195-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-196-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-218-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-22-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-220-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-233-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-248-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-249-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-251-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-252-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-253-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-256-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-275-real true	true ✓	no verdict	no verdict	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-276-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-283-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-286-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-289-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-31-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-326-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-33-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-35-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-354-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-356-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-359-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	true ✓
sp-acj4-364-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-367-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-368-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-369-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-370-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-371-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-372-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-373-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-38-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-394-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-395-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-396-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-397-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-402-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-404-real true	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-412-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-413-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-414-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-415-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-417-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-421-real true	no verdict	no verdict	no verdict	no verdict	true ✓	true ✓	no verdict	no verdict
sp-acj4-5-real contingent	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict	no verdict
sp-acj4-73-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-80-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-83-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-86-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-90-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓
sp-acj4-99-real true	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓	true ✓

sp-acj4-1-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \arcsin\left(\left|x\right|\right) \overset{?}{=} \left|\arcsin x\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0 -> abs asin _0

Lean proof: included

(case sp-acj4-1-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[1] + rules.json[0]: asin abs _0  ->  abs asin _0")
  (var x real)
  (ask (= (asin (abs x)) (abs (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r1w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-102-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{6} = \left|\frac{x}{-6}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6 -> abs / _0 (-6)

Lean proof: included

(case sp-acj4-102-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[102] + rules.json[81]: / abs _0 6  ->  abs / _0 (-6)")
  (var x real)
  (ask (= (/ (abs x) 6) (abs (/ x -6))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-108-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{\left|x\right|}{x} = \frac{x}{\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0 -> / _0 abs _0

Lean proof: included

(case sp-acj4-108-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[108] + rules.json[87]: / abs _0 _0  ->  / _0 abs _0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (abs x)))
  (ask (= (/ (abs x) x) (/ x (abs x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; have hr : (r:ℂ) ≠ 0 := h0.2; have ha : ((|r|:ℝ):ℂ) ≠ 0 := (by have := h1.2; rwa [AE.abs_ofReal] at this); simp only [AE.abs_ofReal, AE.inv_def]; rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_eq_div_iff hr ha]; exact_mod_cast (abs_mul_abs_self r)"))

sp-acj4-120-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{\pi}{\left|x\right|} = \left|\frac{\pi}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0 -> abs / np.pi _0

Lean proof: included

(case sp-acj4-120-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[120] + rules.json[92]: / np.pi abs _0  ->  abs / np.pi _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ pi (abs x)) (abs (/ pi x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, abs_of_pos Real.pi_pos]"))

sp-acj4-127-real

Given

\(x \in \mathbb{R},\quad e^{x} \neq 1,\quad e^{x} \neq -1\)

Claim

\(\displaystyle \left|\operatorname{artanh}\left(e^{x}\right)\right| \overset{?}{=} \operatorname{artanh}\left(e^{x}\right)\)

Expected

contingent true at \(x = -1\); false at \(x = 1\)

equal at some admissible points, different at others

witness (true): \(x = -1\)

witness (false): \(x = 1\)

SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0 -> atanh exp _0

Lean proof: included

(case sp-acj4-127-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[127] + rules.json[96]: abs atanh exp _0  ->  atanh exp _0")
  (var x real)
  (assume (!= (exp x) 1))
  (assume (!= (exp x) -1))
  (ask (= (abs (atanh (exp x))) (atanh (exp x))))
  (expect contingent)
  (witness true (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w1 | exact AE.w2_exp_neg_one_ne_one | exact AE.w2_exp_neg_one_ne_neg_one | exact AE.w2_atanhD_exp_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r127w2 | exact AE.w2_exp_one_ne_one | exact AE.w2_exp_one_ne_neg_one | exact AE.w2_atanhD_exp_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-128-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\cos\left(\arcsin x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0 -> cos asin _0

Lean proof: included

(case sp-acj4-128-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[128] + rules.json[97]: abs cos asin _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (cos (asin x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r128w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-131-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\cosh x\right)\right| = \log\left(\cosh x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0 -> log cosh _0

Lean proof: included

(case sp-acj4-131-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[131] + rules.json[99]: abs log cosh _0  ->  log cosh _0")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (abs (log (cosh x))) (log (cosh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h0.2⟩, by simp⟩, trivial, h0.2⟩, ?_⟩; rw [AE.cosh_ofReal, AE.log_ofReal (Real.cosh_pos r).le, AE.abs_ofReal, abs_of_nonneg (Real.log_nonneg (Real.one_le_cosh r))]"))

sp-acj4-132-real

Given

\(x \in \mathbb{R},\quad \sin x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\sin x\right)\right| \overset{?}{=} -\log\left(\sin x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0 -> neg log sin _0

Lean proof: included

(case sp-acj4-132-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[132] + rules.json[100]: abs log sin _0  ->  neg log sin _0")
  (var x real)
  (assume (nonzero (sin x)))
  (ask (= (abs (log (sin x))) (- (log (sin x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w1 | exact AE.w2_sin_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r132w2 | exact AE.w2_sin_neg_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-133-real

Given

\(x \in \mathbb{R},\quad \tanh x \neq 0\)

Claim

\(\displaystyle \left|\log\left(\tanh x\right)\right| \overset{?}{=} -\log\left(\tanh x\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0 -> neg log tanh _0

Lean proof: included

(case sp-acj4-133-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[133] + rules.json[101]: abs log tanh _0  ->  neg log tanh _0")
  (var x real)
  (assume (nonzero (tanh x)))
  (ask (= (abs (log (tanh x))) (- (log (tanh x)))))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w1 | exact AE.w2_tanh_one_ne | exact AE.w2_tanhD_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r133w2 | exact AE.w2_tanh_neg_one_ne | exact AE.w2_tanhD_neg_one | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-136-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(e^{1}\right) > 0\)

Claim

\(\displaystyle \left|x^{e^{1}}\right| \overset{?}{=} x^{e^{1}}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e -> pow _0 np.e

Lean proof: included

(case sp-acj4-136-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[136] + rules.json[104]: abs pow _0 np.e  ->  pow _0 np.e")
  (var x real)
  (assume (or (nonzero x) (> (re (exp 1)) 0)))
  (ask (= (abs (pow x (exp 1))) (pow x (exp 1))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first  | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, Complex.exp_im]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r136w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-137-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)

Claim

\(\displaystyle \left|x^{\pi}\right| \overset{?}{=} x^{\pi}\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi -> pow _0 np.pi

Lean proof: included

(case sp-acj4-137-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[137] + rules.json[105]: abs pow _0 np.pi  ->  pow _0 np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (ask (= (abs (pow x pi)) (pow x pi)))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r137w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-14-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\sin x = \sin\left(-x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0 -> sin neg _0

Lean proof: included

(case sp-acj4-14-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[14] + rules.json[10]: neg sin _0  ->  sin neg _0")
  (var x real)
  (ask (= (- (sin x)) (sin (- x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [Complex.sin_neg]"))

sp-acj4-145-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|\sin\left(\arccos x\right)\right| \overset{?}{=} \cos\left(\arcsin x\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0 -> cos asin _0

Lean proof: included

(case sp-acj4-145-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0  ->  cos asin _0")
  (var x real)
  (ask (= (abs (sin (acos x))) (cos (asin x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r145w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-162-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{arcosh}\left(\operatorname{surd}\left(x, 5\right)\right) \overset{?}{=} \operatorname{arcosh}\left(x^{\frac{1}{5}}\right)\)

Expected

contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5 -> acosh pow _0 0.2

Lean proof: included

(case sp-acj4-162-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5  ->  acosh pow _0 0.2")
  (var x real)
  (ask (= (acosh (surd x 5)) (acosh (pow x 1/5))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r162w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-179-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arccos x\right) \neq i,\quad \sinh\left(\arccos x\right) \neq -i,\quad \cosh\left(\arccos x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arccos x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arccos x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0 -> asin tanh acos _0

Lean proof: included

(case sp-acj4-179-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[179] + rules.json[128]: atan sinh acos _0  ->  asin tanh acos _0")
  (var x real)
  (assume (!= (sinh (acos x)) i))
  (assume (!= (sinh (acos x)) (- i)))
  (assume (nonzero (cosh (acos x))))
  (ask (= (atan (sinh (acos x))) (asin (tanh (acos x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w1 | exact AE.w2_r179w1_hI | exact AE.w2_r179w1_hnI | exact AE.w2_r179w1_atanD | exact AE.w2_r179w1_cosh | exact AE.w2_r179w1_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r179w2 | exact AE.w2_r179_hI | exact AE.w2_r179_hnI | exact AE.w2_r179_atanD | exact AE.w2_r179_cosh | exact AE.w2_r179_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-181-real

Given

\(x \in \mathbb{R},\quad \sinh\left(\arcsin x\right) \neq i,\quad \sinh\left(\arcsin x\right) \neq -i,\quad \cosh\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \arctan\left(\sinh\left(\arcsin x\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\arcsin x\right)\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 3\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 3\)

SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0 -> asin tanh asin _0

Lean proof: included

(case sp-acj4-181-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[181] + rules.json[129]: atan sinh asin _0  ->  asin tanh asin _0")
  (var x real)
  (assume (!= (sinh (asin x)) i))
  (assume (!= (sinh (asin x)) (- i)))
  (assume (nonzero (cosh (asin x))))
  (ask (= (atan (sinh (asin x))) (asin (tanh (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 3) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r181w2 | exact AE.w2_r181_hI | exact AE.w2_r181_hnI | exact AE.w2_r181_atanD | exact AE.w2_r181_cosh | exact AE.w2_r181_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-185-real

Given

\(x \in \mathbb{R},\quad \cos\left(\cos x\right) \neq 0,\quad \tan\left(\cos x\right) \neq i,\quad \tan\left(\cos x\right) \neq -i\)

Claim

\(\displaystyle \arctan\left(\tan\left(\cos x\right)\right) = \cos x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0 -> cos _0

Lean proof: included

(case sp-acj4-185-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[185] + rules.json[132]: atan tan cos _0  ->  cos _0")
  (var x real)
  (assume (nonzero (cos (cos x))))
  (assume (!= (tan (cos x)) i))
  (assume (!= (tan (cos x)) (- i)))
  (ask (= (atan (tan (cos x))) (cos x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨⟨trivial, h1.1.2⟩, by rw [AE.cos_ofReal, AE.tan_ofReal]; exact AE.atan.D_ofReal _⟩, trivial⟩, ?_⟩; rw [AE.cos_ofReal, AE.tan_ofReal, AE.atan_ofReal, Real.arctan_tan (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]) (by linarith [Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-194-real

Given

\(x \in \mathbb{R},\quad \sin\left(\cos x\right) \neq 1,\quad \sin\left(\cos x\right) \neq -1,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\cos x\right)\right) = \operatorname{arsinh}\left(\tan\left(\cos x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0 -> asinh tan cos _0

Lean proof: included

(case sp-acj4-194-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[194] + rules.json[136]: atanh sin cos _0  ->  asinh tan cos _0")
  (var x real)
  (assume (!= (sin (cos x)) 1))
  (assume (!= (sin (cos x)) -1))
  (assume (nonzero (cos (cos x))))
  (ask (= (atanh (sin (cos x))) (asinh (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.cos r := (by linarith [Real.neg_one_le_cos r, Real.pi_gt_three]); have hy2 : Real.cos r < Real.pi / 2 := (by linarith [Real.cos_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.cos_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.cos r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.cos r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.cos_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-195-real

Given

\(x \in \mathbb{R},\quad \sin\left(\sin x\right) \neq 1,\quad \sin\left(\sin x\right) \neq -1,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\sin x\right)\right) = \operatorname{arsinh}\left(\tan\left(\sin x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0 -> asinh tan sin _0

Lean proof: included

(case sp-acj4-195-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[195] + rules.json[137]: atanh sin sin _0  ->  asinh tan sin _0")
  (var x real)
  (assume (!= (sin (sin x)) 1))
  (assume (!= (sin (sin x)) -1))
  (assume (nonzero (cos (sin x))))
  (ask (= (atanh (sin (sin x))) (asinh (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.sin r := (by linarith [Real.neg_one_le_sin r, Real.pi_gt_three]); have hy2 : Real.sin r < Real.pi / 2 := (by linarith [Real.sin_le_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.sin_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.sin r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.sin r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp, by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp, by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp⟩, ?_⟩; rw [AE.sin_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-196-real

Given

\(x \in \mathbb{R},\quad \sin\left(\tanh x\right) \neq 1,\quad \sin\left(\tanh x\right) \neq -1,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\tanh x\right)\right) = \operatorname{arsinh}\left(\tan\left(\tanh x\right)\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0 -> asinh tan tanh _0

Lean proof: included

(case sp-acj4-196-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[196] + rules.json[138]: atanh sin tanh _0  ->  asinh tan tanh _0")
  (var x real)
  (assume (!= (sin (tanh x)) 1))
  (assume (!= (sin (tanh x)) -1))
  (assume (nonzero (cos (tanh x))))
  (ask (= (atanh (sin (tanh x))) (asinh (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hy1 : -(Real.pi / 2) < Real.tanh r := (by linarith [Real.neg_one_lt_tanh r, Real.pi_gt_three]); have hy2 : Real.tanh r < Real.pi / 2 := (by linarith [Real.tanh_lt_one r, Real.pi_gt_three]); have ha := h0.2; have hb := h1.2; rw [AE.tanh_ofReal, AE.sin_ofReal] at ha hb; have ha' : Real.sin (Real.tanh r) ≠ 1 := (by exact_mod_cast ha); have hb' : Real.sin (Real.tanh r) ≠ -1 := (by exact_mod_cast hb); refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.atanh.D_iff]; exact ⟨h0.2, h1.2⟩⟩, ⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h2.2⟩, by simp [AE.cosh_ofReal_ne_zero]⟩, ?_⟩; rw [AE.tanh_ofReal, AE.sin_ofReal, AE.tan_ofReal, AE.atanh_ofReal (lt_of_le_of_ne (Real.neg_one_le_sin _) hb'.symm) (lt_of_le_of_ne (Real.sin_le_one _) ha'), AE.asinh_ofReal, AE.artanh_sin_eq_arsinh_tan hy1 hy2]"))

sp-acj4-218-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-6}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-6}{x}} = \frac{x}{-6}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0 -> / !0 (-6)

Lean proof: included

(case sp-acj4-218-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[218] + rules.json[150]: inv / (-6) !0  ->  / !0 (-6)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -6 x)))
  (ask (= (/ (/ -6 x)) (/ x -6)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-22-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \tanh\left(\left|x\right|\right) = \left|\tanh x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _0

Lean proof: included

(case sp-acj4-22-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0  ->  abs tanh _0")
  (var x real)
  (ask (= (tanh (abs x)) (abs (tanh x))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [AE.abs_ofReal, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal]; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.tanh_abs]"))

sp-acj4-220-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \frac{-8}{x} \neq 0\)

Claim

\(\displaystyle \frac{1}{\frac{-8}{x}} = \frac{x}{-8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0 -> / !0 (-8)

Lean proof: included

(case sp-acj4-220-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[220] + rules.json[152]: inv / (-8) !0  ->  / !0 (-8)")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (/ -8 x)))
  (ask (= (/ (/ -8 x)) (/ x -8)))
  (expect true)
  (lean-proof "refine ⟨⟨h0.2, h1.2⟩, ?_⟩; have hx : x ≠ 0 := h0.2; simp only [AE.inv_def]; field_simp; try ring"))

sp-acj4-233-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0,\quad x^{\pi} \neq 0,\quad x \neq 0 \lor \operatorname{Re}\left(-\pi\right) > 0\)

Claim

\(\displaystyle \frac{1}{x^{\pi}} = x^{-\pi}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi -> pow _0 neg np.pi

Lean proof: included

(case sp-acj4-233-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[233] + rules.json[162]: inv pow _0 np.pi  ->  pow _0 neg np.pi")
  (var x real)
  (assume (or (nonzero x) (> (re pi) 0)))
  (assume (nonzero (pow x pi)))
  (assume (or (nonzero x) (> (re (- pi)) 0)))
  (ask (= (/ (pow x pi)) (pow x (- pi))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := (by intro h; apply h1.2; rw [h]; simp); refine ⟨⟨⟨by simp [hx], h1.2⟩, by simp [hx]⟩, ?_⟩; simp only [AE.pow, if_neg hx, AE.inv_def, AE.exp_def]; rw [← Complex.exp_neg]; ring_nf"))

sp-acj4-248-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\arcsin x\right| \overset{?}{=} \arcsin\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0 -> asin neg abs _0

Lean proof: included

(case sp-acj4-248-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[248] + rules.json[164]: neg abs asin _0  ->  asin neg abs _0")
  (var x real)
  (ask (= (- (abs (asin x))) (asin (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r248w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-249-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\operatorname{arsinh}\left(x\right)\right| = \operatorname{arsinh}\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0 -> asinh neg abs _0

Lean proof: included

(case sp-acj4-249-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[249] + rules.json[165]: neg abs asinh _0  ->  asinh neg abs _0")
  (var x real)
  (ask (= (- (abs (asinh x))) (asinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.asinh_ofReal, AE.asinh_ofReal, AE.abs_ofReal, Real.arsinh_neg, AE.arsinh_abs]; push_cast; ring"))

sp-acj4-251-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1,\quad -\left|x\right| \neq 1,\quad -\left|x\right| \neq -1\)

Claim

\(\displaystyle -\left|\operatorname{artanh}\left(x\right)\right| \overset{?}{=} \operatorname{artanh}\left(-\left|x\right|\right)\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0 -> atanh neg abs _0

Lean proof: included

(case sp-acj4-251-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[251] + rules.json[167]: neg abs atanh _0  ->  atanh neg abs _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (assume (!= (- (abs x)) 1))
  (assume (!= (- (abs x)) -1))
  (ask (= (- (abs (atanh x))) (atanh (- (abs x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r251w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-252-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\sinh x\right| = \sinh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0 -> sinh neg abs _0

Lean proof: included

(case sp-acj4-252-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[252] + rules.json[168]: neg abs sinh _0  ->  sinh neg abs _0")
  (var x real)
  (ask (= (- (abs (sinh x))) (sinh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.sinh_ofReal, AE.sinh_ofReal, AE.abs_ofReal, Real.abs_sinh, Real.sinh_neg]; push_cast; ring"))

sp-acj4-253-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle -\left|\tanh x\right| = \tanh\left(-\left|x\right|\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0 -> tanh neg abs _0

Lean proof: included

(case sp-acj4-253-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[253] + rules.json[169]: neg abs tanh _0  ->  tanh neg abs _0")
  (var x real)
  (ask (= (- (abs (tanh x))) (tanh (- (abs x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; rw [AE.abs_ofReal r, ← Complex.ofReal_neg |r|, AE.tanh_ofReal, AE.tanh_ofReal, AE.abs_ofReal, ← AE.tanh_abs, Real.tanh_neg]; push_cast; ring"))

sp-acj4-256-real

Given

\(x \in \mathbb{R},\quad -e^{x} \neq 0\)

Claim

\(\displaystyle -e^{-x} = \frac{1}{-e^{x}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0 -> inv neg exp !0

Lean proof: included

(case sp-acj4-256-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[256] + rules.json[171]: neg exp neg !0  ->  inv neg exp !0")
  (var x real)
  (assume (nonzero (- (exp x))))
  (ask (= (- (exp (- x))) (/ (- (exp x)))))
  (expect true)
  (lean-proof "refine ⟨h0.2, ?_⟩; simp [Complex.exp_neg]"))

sp-acj4-275-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-3} = \left|x^{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3) -> abs pow _0 (-3)

Lean proof: included

(case sp-acj4-275-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[275] + rules.json[184]: pow abs _0 (-3)  ->  abs pow _0 (-3)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -3) (abs (pow x -3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr], by simp⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_inv, AE.abs_eq, AE.abs_eq]; simp"))

sp-acj4-276-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \left|x\right|^{-4} = x^{-4}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4) -> pow _0 (-4)

Lean proof: included

(case sp-acj4-276-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[276] + rules.json[185]: pow abs _0 (-4)  ->  pow _0 (-4)")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (pow (abs x) -4) (pow x -4)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : (r:ℂ) ≠ 0 := h1.2; have ha : AE.abs (r:ℂ) ≠ 0 := h0.2; refine ⟨⟨⟨by simp, by simp [ha]⟩, by simp [hr]⟩, ?_⟩; rw [AE.pow_neg_ofNat ha, AE.pow_neg_ofNat hr, AE.abs_ofReal]; congr 1; exact_mod_cast (Even.pow_abs ⟨2, by norm_num⟩ r)"))

sp-acj4-283-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{2} = x \, x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2 -> * _0 _0

Lean proof: included

(case sp-acj4-283-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[283] + rules.json[192]: pow abs _0 2  ->  * _0 _0")
  (var x real)
  (ask (= (pow (abs x) 2) (* x x)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (by rw [sq_abs]; ring : |r|^2 = r*r)"))

sp-acj4-286-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{5} = \left|x^{5}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5 -> abs pow _0 5

Lean proof: included

(case sp-acj4-286-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[286] + rules.json[195]: pow abs _0 5  ->  abs pow _0 5")
  (var x real)
  (ask (= (pow (abs x) 5) (abs (pow x 5))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal r, ← Complex.ofReal_pow r, AE.abs_ofReal (r^5), abs_pow]; push_cast; ring"))

sp-acj4-289-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left|x\right|^{8} = x^{8}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8 -> pow _0 8

Lean proof: included

(case sp-acj4-289-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[289] + rules.json[198]: pow abs _0 8  ->  pow _0 8")
  (var x real)
  (ask (= (pow (abs x) 8) (pow x 8)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.pow_ofNat_real, AE.abs_ofReal]; exact_mod_cast (Even.pow_abs ⟨4, by norm_num⟩ r)"))

sp-acj4-31-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 5 \, \left|x\right| = \left|-5 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0 -> abs * (-5) _0

Lean proof: included

(case sp-acj4-31-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[31] + rules.json[24]: * 5 abs _0  ->  abs * (-5) _0")
  (var x real)
  (ask (= (* 5 (abs x)) (abs (* -5 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-326-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \left(e^{x}\right)^{2} = e^{2 \, x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2 -> exp * 2 _0

Lean proof: included

(case sp-acj4-326-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[326] + rules.json[208]: pow exp _0 2  ->  exp * 2 _0")
  (var x real)
  (ask (= (pow (exp x) 2) (exp (* 2 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; rw [AE.pow_ofNat_real, AE.exp_def, AE.exp_def, ← Complex.exp_nat_mul]; norm_num"))

sp-acj4-33-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle 6 \, \left|x\right| = \left|-6 \, x\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0 -> abs * (-6) _0

Lean proof: included

(case sp-acj4-33-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[33] + rules.json[26]: * 6 abs _0  ->  abs * (-6) _0")
  (var x real)
  (ask (= (* 6 (abs x)) (abs (* -6 x))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq]"))

sp-acj4-35-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{7} \neq 0\)

Claim

\(\displaystyle 7 \, \log x \overset{?}{=} \log\left(x^{7}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0 -> log pow _0 7

Lean proof: included

(case sp-acj4-35-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[35] + rules.json[28]: * 7 log _0  ->  log pow _0 7")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 7)))
  (ask (= (* 7 (log x)) (log (pow x 7))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r35w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-354-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 5\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 5} = 5^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5 -> pow 5 log _0

Lean proof: included

(case sp-acj4-354-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[354] + rules.json[224]: pow _0 log 5  ->  pow 5 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 5)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 5)) (pow 5 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-356-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log 7\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log 7} = 7^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7 -> pow 7 log _0

Lean proof: included

(case sp-acj4-356-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[356] + rules.json[226]: pow _0 log 7  ->  pow 7 log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log 7)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log 7)) (pow 7 (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by norm_num, by simp [hx]⟩, hx, by simp⟩, ?_⟩; simp [AE.pow, hx, mul_comm]"))

sp-acj4-359-real

Given

\(x \in \mathbb{R},\quad x \neq 0 \lor \operatorname{Re}\left(\log \pi\right) > 0,\quad x \neq 0\)

Claim

\(\displaystyle x^{\log \pi} = \pi^{\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi -> pow np.pi log _0

Lean proof: included

(case sp-acj4-359-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[359] + rules.json[227]: pow _0 log np.pi  ->  pow np.pi log _0")
  (var x real)
  (assume (or (nonzero x) (> (re (log pi)) 0)))
  (assume (nonzero x))
  (ask (= (pow x (log pi)) (pow pi (log x))))
  (expect true)
  (lean-proof "have hx : x ≠ 0 := h1.2; refine ⟨⟨⟨by simp [Real.pi_ne_zero], by simp [hx]⟩, hx, by simp [Real.pi_ne_zero]⟩, ?_⟩; simp [AE.pow, hx, Real.pi_ne_zero, mul_comm]"))

sp-acj4-364-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 3\right) = \left|\operatorname{surd}\left(x, 3\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3 -> abs rootn _0 3

Lean proof: included

(case sp-acj4-364-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[364] + rules.json[232]: rootn abs _0 3  ->  abs rootn _0 3")
  (var x real)
  (ask (= (surd (abs x) 3) (abs (surd x 3))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (3:ℂ) = ((3:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 3, by norm_num, Or.inr (AE.odd_ofInt ⟨1, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-367-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\left|x\right|, 9\right) = \left|\operatorname{surd}\left(x, 9\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9 -> abs rootn _0 9

Lean proof: included

(case sp-acj4-367-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[367] + rules.json[235]: rootn abs _0 9  ->  abs rootn _0 9")
  (var x real)
  (ask (= (surd (abs x) 9) (abs (surd x 9))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; rw [show (9:ℂ) = ((9:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨AE.abs_isReal _, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩⟩, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt 9, by norm_num, Or.inr (AE.odd_ofInt ⟨4, by norm_num⟩), Or.inr (by norm_num)⟩, by simp⟩, ?_⟩; rw [AE.abs_ofReal, AE.surd_abs_odd r (by norm_num)]"))

sp-acj4-368-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, -5\right) = \left(\arccos x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5) -> pow acos _0 -0.2

Lean proof: included

(case sp-acj4-368-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5)  ->  pow acos _0 -0.2")
  (var x real)
  (assume (real (acos x)))
  (assume (nonzero (acos x)))
  (ask (= (surd (acos x) -5) (pow (acos x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; have hne : Real.arccos r ≠ 0 := (by have := h1.2; rw [AE.acos_ofReal hI.1 hI.2] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-369-real

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2

Lean proof: included

(case sp-acj4-369-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5  ->  pow acos _0 0.2")
  (var x real)
  (assume (real (acos x)))
  (ask (= (surd (acos x) 5) (pow (acos x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-370-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2

Lean proof: included

(case sp-acj4-370-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5)  ->  pow acosh _0 -0.2")
  (var x real)
  (assume (real (acosh x)))
  (assume (nonzero (acosh x)))
  (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; have hne : Real.arcosh r ≠ 0 := (by have := h1.2; rw [AE.acosh_ofReal hI] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))

sp-acj4-371-real

Given

\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), 5\right) = \operatorname{arcosh}\left(x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5 -> pow acosh _0 0.2

Lean proof: included

(case sp-acj4-371-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[371] + rules.json[239]: rootn acosh _0 5  ->  pow acosh _0 0.2")
  (var x real)
  (assume (real (acosh x)))
  (ask (= (surd (acosh x) 5) (pow (acosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; rw [show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inr (by norm_num)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inr (by norm_num)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inr (by norm_num))]"))

sp-acj4-372-real

Given

\(x \in \mathbb{R},\quad \cosh x \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, -5\right) = \left(\cosh x\right)^{-\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _0 -0.2

Lean proof: included

(case sp-acj4-372-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5)  ->  pow cosh _0 -0.2")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-373-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\cosh x, 5\right) = \left(\cosh x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2

Lean proof: included

(case sp-acj4-373-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5  ->  pow cosh _0 0.2")
  (var x real)
  (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show ((1:ℂ)/5) = (((((5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (5:ℂ) = ((5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), AE.pow_ofReal hc.le (Or.inl hc.ne')]"))

sp-acj4-38-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad x^{9} \neq 0\)

Claim

\(\displaystyle 9 \, \log x \overset{?}{=} \log\left(x^{9}\right)\)

Expected

contingent true at \(x = 1\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0 -> log pow _0 9

Lean proof: included

(case sp-acj4-38-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[38] + rules.json[31]: * 9 log _0  ->  log pow _0 9")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (pow x 9)))
  (ask (= (* 9 (log x)) (log (pow x 9))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r38w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-394-real

Given

\(x \in \mathbb{R},\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\left|\operatorname{artanh}\left(x\right)\right|\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-394-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[394] + rules.json[242]: sin abs atanh _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (abs (atanh x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r394w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-395-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\cos x\right|\right) = \left|\sin\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0 -> abs sin cos _0

Lean proof: included

(case sp-acj4-395-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[395] + rules.json[243]: sin abs cos _0  ->  abs sin cos _0")
  (var x real)
  (ask (= (sin (abs (cos x))) (abs (sin (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.cos_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-396-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\sin x\right|\right) = \left|\sin\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0 -> abs sin sin _0

Lean proof: included

(case sp-acj4-396-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[396] + rules.json[244]: sin abs sin _0  ->  abs sin sin _0")
  (var x real)
  (ask (= (sin (abs (sin x))) (abs (sin (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.sin_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-397-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\left|\tanh x\right|\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-397-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[397] + rules.json[245]: sin abs tanh _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (abs (tanh x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-402-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \sin\left(\operatorname{artanh}\left(\left|x\right|\right)\right) \overset{?}{=} \left|\sin\left(\operatorname{artanh}\left(x\right)\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0 -> abs sin atanh _0

Lean proof: included

(case sp-acj4-402-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[402] + rules.json[246]: sin atanh abs _0  ->  abs sin atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (sin (atanh (abs x))) (abs (sin (atanh x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r402w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-404-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \sin\left(\tanh\left(\left|x\right|\right)\right) = \left|\sin\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0 -> abs sin tanh _0

Lean proof: included

(case sp-acj4-404-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[404] + rules.json[247]: sin tanh abs _0  ->  abs sin tanh _0")
  (var x real)
  (ask (= (sin (tanh (abs x))) (abs (sin (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], ?_⟩; simp only [AE.tanh_ofReal, AE.sin_ofReal, AE.abs_ofReal]; rw [AE.tanh_abs, AE.sin_abs_of_le_pi (by rw [abs_le]; constructor <;> linarith [(Real.neg_one_lt_tanh r).le, (Real.tanh_lt_one r).le, Real.pi_gt_three])]"))

sp-acj4-412-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\arcsin x\right|\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\arcsin x\right|\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-412-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (abs (asin x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (abs (asin x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r412w2 | exact AE.w2_r412_cos | exact AE.w2_r412_tanD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-413-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\cos x\right|\right) \neq 0,\quad \cos\left(\cos x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\cos x\right|\right) = \left|\tan\left(\cos x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0 -> abs tan cos _0

Lean proof: included

(case sp-acj4-413-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[413] + rules.json[254]: tan abs cos _0  ->  abs tan cos _0")
  (var x real)
  (assume (nonzero (cos (abs (cos x)))))
  (assume (nonzero (cos (cos x))))
  (ask (= (tan (abs (cos x))) (abs (tan (cos x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.cos_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_cos r, Real.cos_le_one r, Real.pi_gt_three])]"))

sp-acj4-414-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\sin x\right|\right) \neq 0,\quad \cos\left(\sin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\sin x\right|\right) = \left|\tan\left(\sin x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0 -> abs tan sin _0

Lean proof: included

(case sp-acj4-414-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[414] + rules.json[255]: tan abs sin _0  ->  abs tan sin _0")
  (var x real)
  (assume (nonzero (cos (abs (sin x)))))
  (assume (nonzero (cos (sin x))))
  (ask (= (tan (abs (sin x))) (abs (tan (sin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.sin_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_le_sin r, Real.sin_le_one r, Real.pi_gt_three])]"))

sp-acj4-415-real

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\tanh x\right|\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\tanh x\right|\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-415-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[415] + rules.json[256]: tan abs tanh _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (abs (tanh x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (abs (tanh x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.tanh_ofReal, AE.abs_ofReal, AE.tan_ofReal]; rw [AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-417-real

Given

\(x \in \mathbb{R},\quad \cos\left(\arcsin\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\arcsin\left(\left|x\right|\right)\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0 -> abs tan asin _0

Lean proof: included

(case sp-acj4-417-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (asin (abs x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (asin (abs x))) (abs (tan (asin x)))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r417w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_cos_asin_two]")))

sp-acj4-421-real

Given

\(x \in \mathbb{R},\quad \cos\left(\tanh\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\tanh x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\tanh\left(\left|x\right|\right)\right) = \left|\tan\left(\tanh x\right)\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0 -> abs tan tanh _0

Lean proof: included

(case sp-acj4-421-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[421] + rules.json[260]: tan tanh abs _0  ->  abs tan tanh _0")
  (var x real)
  (assume (nonzero (cos (tanh (abs x)))))
  (assume (nonzero (cos (tanh x))))
  (ask (= (tan (tanh (abs x))) (abs (tan (tanh x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨⟨⟨by simp [AE.abs_ofReal, AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h0.2⟩, ⟨⟨by simp [AE.cosh_ofReal_ne_zero], by rw [AE.tan.D_iff, ← AE.cos_eq]; exact h1.2⟩, by simp⟩⟩, ?_⟩; simp only [AE.abs_ofReal, AE.tanh_ofReal, AE.tan_ofReal]; rw [AE.tanh_abs, AE.tan_abs_of_lt (by rw [abs_lt]; constructor <;> linarith [Real.neg_one_lt_tanh r, Real.tanh_lt_one r, Real.pi_gt_three])]"))

sp-acj4-5-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 1,\quad x \neq 1,\quad x \neq -1\)

Claim

\(\displaystyle \operatorname{artanh}\left(\left|x\right|\right) \overset{?}{=} \left|\operatorname{artanh}\left(x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0 -> abs atanh _0

Lean proof: included

(case sp-acj4-5-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[5] + rules.json[3]: atanh abs _0  ->  abs atanh _0")
  (var x real)
  (assume (!= (abs x) 1))
  (assume (!= x 1))
  (assume (!= x -1))
  (ask (= (atanh (abs x)) (abs (atanh x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]"))
  (witness false (x 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r5w2 | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))

sp-acj4-73-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad -\left|x\right| \neq 0\)

Claim

\(\displaystyle \frac{-1}{\left|x\right|} = \frac{1}{-\left|x\right|}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0 -> inv neg abs !0

Lean proof: included

(case sp-acj4-73-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[73] + rules.json[52]: / (-1) abs !0  ->  inv neg abs !0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero (- (abs x))))
  (ask (= (/ -1 (abs x)) (/ (- (abs x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-80-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad -\cos x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\cos x} = \frac{1}{-\cos x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0 -> inv neg cos _0

Lean proof: included

(case sp-acj4-80-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[80] + rules.json[59]: / (-1) cos _0  ->  inv neg cos _0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (- (cos x))))
  (ask (= (/ -1 (cos x)) (/ (- (cos x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, by simp, h1.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-83-real

Given

\(x \in \mathbb{R},\quad x \neq 0,\quad \log x \neq 0,\quad -\log x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\log x} = \frac{1}{-\log x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0 -> inv neg log !0

Lean proof: included

(case sp-acj4-83-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[83] + rules.json[62]: / (-1) log !0  ->  inv neg log !0")
  (var x real)
  (assume (nonzero x))
  (assume (nonzero (log x)))
  (assume (nonzero (- (log x))))
  (ask (= (/ -1 (log x)) (/ (- (log x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h0.2, h1.2⟩, h0.2, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-86-real

Given

\(x \in \mathbb{R},\quad \cos x \neq 0,\quad \tan x \neq 0,\quad -\tan x \neq 0\)

Claim

\(\displaystyle \frac{-1}{\tan x} = \frac{1}{-\tan x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0 -> inv neg tan !0

Lean proof: included

(case sp-acj4-86-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[86] + rules.json[65]: / (-1) tan !0  ->  inv neg tan !0")
  (var x real)
  (assume (nonzero (cos x)))
  (assume (nonzero (tan x)))
  (assume (nonzero (- (tan x))))
  (ask (= (/ -1 (tan x)) (/ (- (tan x)))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨h1.1.1, h1.2⟩, h1.1.1, h2.2⟩, ?_⟩; simp only [AE.inv_def]; rw [inv_neg]; ring"))

sp-acj4-90-real

Given

\(x \in \mathbb{R},\quad \left|x\right| \neq 0,\quad x \neq 0\)

Claim

\(\displaystyle \frac{3}{\left|x\right|} = \left|\frac{-3}{x}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0 -> abs / (-3) _0

Lean proof: included

(case sp-acj4-90-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[90] + rules.json[69]: / 3 abs _0  ->  abs / (-3) _0")
  (var x real)
  (assume (nonzero (abs x)))
  (assume (nonzero x))
  (ask (= (/ 3 (abs x)) (abs (/ -3 x))))
  (expect true)
  (lean-proof "refine ⟨⟨⟨by simp, h0.2⟩, h1.2, by simp⟩, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

sp-acj4-99-real

Given

\(x \in \mathbb{R}\)

Claim

\(\displaystyle \frac{\left|x\right|}{3} = \left|\frac{x}{-3}\right|\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3 -> abs / _0 (-3)

Lean proof: included

(case sp-acj4-99-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[99] + rules.json[78]: / abs _0 3  ->  abs / _0 (-3)")
  (var x real)
  (ask (= (/ (abs x) 3) (abs (/ x -3))))
  (expect true)
  (lean-proof "refine ⟨by simp, ?_⟩; simp [AE.abs_eq, div_eq_mul_inv]"))

simplipy-real

Case set ba9dc4def822

Case set 3ba0dfac4bca

Case set ee702ae2fe06

Case set `ba9dc4def822`

Case set `3ba0dfac4bca`

Case set `ee702ae2fe06`